Periodic solutions of inhomogeneous Schrödinger flows into 2-sphere
Pingliang Huang, Youde Wang · Discrete and Continuous Dynamical Systems - S · 2016
In this paper,we consider the so called generalized inhomogeneousSchrödinger flows from a closed Riemann surface $M$ into thestandard 2-sphere $S^2$ associated with the energy functional givenby\begin{align*}E_{f,P}(u)=\int_M\left(\frac{1}{2}f| abla u|^2+P(u_3)\right)dV_g.\end{align*}We showed the existence of special periodic solutions to thegeneralized inhomogeneous Schrödinger flows from $M$ withconvolution symmetry (especially $M = S^2$) into $S^2$ when thefunction $f$ and $P$ satisfy certain conditions respectively.Especially, we show that the inhomogeneous Heisenberg spin chainsystem from a closed Riemann surface with convolution symmetryadmits some special periodic solutions if the couplingfunction $f$ satisfies some suitable conditions. We also prove thatthere exist an infinite number of special periodic solutions to theLandau-Lifshitz system with an external magnetic field from $S^2$into $S^2$.